Crystalline ground states for classical particles

Abstract

Pair interactions whose Fourier transform is nonnegative and vanishes above a wave number K_0 are shown to give rise to periodic and aperiodic infinite volume ground state configurations (GSCs) in any dimension d. A typical three dimensional example is an interaction of asymptotic form cos(K_0 r)/r^4. The result is obtained for densities rho >= rho_d where rho_1=K_0/2pi, rho_2=(sqrt{3}/8)(K_0/pi)^2 and rho_3=(1/8sqrt{2})(K_0/pi)^3. At rho_d there is a unique periodic GSC which is the uniform chain, the triangular lattice and the bcc lattice for d=1,2,3, respectively. For rho>rho_d the GSC is nonunique and the degeneracy is continuous: Any periodic configuration of density rho with all reciprocal lattice vectors not smaller than K_0, and any union of such configurations, is a GSC. The fcc lattice is a GSC only for rho>=(1/6 sqrt{3})(K_0/pi)^3.

final version

Cited by in corpus (36)

• Hyperuniform States of Matter
• Why do ultrasoft repulsive particles cluster and crystallize? Analytical results from density functional theory
• Classical Disordered Ground States: Super-Ideal Gases, and Stealth and Equi-Luminous Materials
• Ensemble Theory for Stealthy Hyperuniform Disordered Ground States
• Phase diagram of Hertzian spheres
• Ground states of stealthy hyperuniform potentials: I. Entropically favored configurations
• Collective Coordinate Control of Density Distributions
• The Perfect Glass Paradigm: Disordered Hyperuniform Glasses Down to Absolute Zero
• Face-centered cubic crystallization of atomistic configurations
• New Duality Relations for Classical Ground States
• From bcc to fcc: interplay between oscillating long-range and repulsive short-range forces
• Ground states and formal duality relations in the Gaussian core model
• On the Support of Minimizers of Causal Variational Principles
• Phonon dispersions of cluster crystals
• Higher Dimensional Coulomb Gases and Renormalized Energy Functionals
• Minimizing lattice structures for Morse potential energy in two and three dimensions
• Periodic striped ground states in Ising models with competing interactions
• Counterintuitive ground states in soft-core models
• Ground state at high density
• Striped periodic minimizers of a two-dimensional model for martensitic phase transitions
• Sequential Phase Transitions and Transient Structured Fluctuations in Two-Dimensional Systems with a High-Density Kagome Lattice Phase
• Optimal lattice configurations for interacting spatially extended particles
• Inherent Structures for Soft Long-Range Interactions in Two-Dimensional Many-Particle Systems
• Realization of stripes and slabs in two and three dimensions
• Exotic Ground States of Directional Pair Potentials via Collective-Density Variables
• Three-dimensional lattice ground states for Riesz and Lennard-Jones type energies
• Effect of periodic arrays of defects on lattice energy minimizers
• Classical Density Functional Theory: The Local Density Approximation
• Distribution of cracks in a chain of atoms at low temperature
• Variational wave functions for homogenous Bose systems
• Nearest-Neighbor Functions for Disordered Stealthy Hyperuniform Many-Particle Systems
• Froth-like minimizers of a non local free energy functional with competing interactions
• Equidistribution of jellium energy for Coulomb and Riesz Interactions
• 2D Theta Functions and Crystallization among Bravais Lattices
• Positive-density ground states of the Gross-Pitaevskii equation
• A possible mechanism of concurring diagonal and off-diagonal long-range order for soft interactions